Recent studies show that liver function is a predominant predictor for overall survival for patients with intrahepatic cancer and cirrhosis regardless of intervention (Johnson, J C O 2015). There is no robust method to quantify regional liver function from the routine clinical multi-phase MRI with gadoxetic-acid injection. Current techniques, including the deconvolution technique, cannot differentiate contrast agents in hepatocytes from in the blood plasma, the first indicates the liver function (capability of uptake of contrast agents) and the latter is related to the hemodynamics of agents in the blood. The dual-input two-compartment model (Sourbron, Radiology 2012) requires fitting six unknown parameters (5 involving hemodynamics of the liver blood flow) and typically requires acquisition of high-temporal resolution dynamic gadoxetic-acid enhanced (DGAE) MRI. The acquisition of high-temporal resolution DGAE MRI challenges routine workflow at a typical MRI clinic and is time consuming when fitting a long dynamic series of data in the whole liver. Due to temporally sparse-sampling in routine clinical multi-phase MRI with gadoxetic-acid injection, the arterial (Ca) and portal vein (Cpv) input functions that are required to fit the dual-input two-compartment model cannot be determined accurately, resulting in a large uncertainty and variation in the fitted parameters. As a result, a semi-quantitative analysis is performed that cannot differentiate intra- and extra-cellular contrasts.